Mixed spectral element method combined with second-order time stepping schemes for a two-dimensional nonlinear fourth-order fractional diffusion equation

Abstract

In this article, a mixed spectral element method combined with second-order time stepping schemes for solving a two-dimensional nonlinear fourth-order fractional diffusion equation is constructed. For formulating an efficient numerical scheme, an auxiliary function is introduced to transform the fourth-order fractional system into a low-order coupled system, then the time direction is discretized by second-order FBT-θ schemes, and the spatial direction is approximated using the Legendre mixed spectral element method (LMSEM). The stability and the optimal error estimate with $O({\tau }^2+h^{\min \{N+1,r\}}{N}^{-r})$ for the fully discrete scheme are derived, where τ stands for the time step size, h denotes the space step size, N indicates the degree of the polynomial, and r represents the order of Sobolev space. Finally, some numerical tests are carried out to verify the theory results and the effectiveness of the developed algorithm.